After
calling attention to the empirical and theoretical motivations for considering
the hypothesis of a self-similar cosmos, the basic concepts and scaling rules
of the Self-Similar Cosmological Model are presented. The results of a diverse
set of 20 falsification tests are then shown to provide strong quantitative
support for the uniqueness and broad applicability of the self-similar scale
transformation equations, which successfully correlate physical parameters of
atomic, stellar, and galactic scale systems. Possible implications of these
results are discussed.

...the wise man looks into space
and does not regard the small as too little,
nor the great as too big,
for he knows that there is no limit to dimensions.Lao-tse

1. INTRODUCTION

1.1. Goals of the Review

The Self-Similar
Cosmological Model (hereafter SSCM) is a heuristic cosmological model that has
been developed over the past 10 years in a series of 17 papers by the author
(Oldershaw, 1978-1987b, 1989a,b). The major goal of this review is to introduce
the SSCM, its 20 successful falsification tests, and its major predictions to
as large and diverse an audience of scientists as possible. Other goals of
this review are a reasonably compact summary of the previous work on the SSCM
and the identification and clarification of various modifications to the model
that have occurred over the past 10 years. The remainder of this section will
be concerned with the question of why one should be interested in an unorthodox
self-similar model of the cosmos. Section 2 will introduce the SSCM in its
simplest and most general form and Section 3 will discuss the substantial amount
of empirical evidence in favor of cosmological self-similarity. The sequel
(Oldershaw, 1989c) to this paper will present a more detailed and technical
discussion of the SSCM, including its major predictions, implications, and unresolved
problems.

1.2. Reasons for Considering a Self-Similar Cosmological
Model

The overwhelming majority of physicists currently think that
the Big Bang cosmology (augmented by Inflation) and the Standard Model of
particle physics (and subsequent unification theories) are unquestionably
the right theories to guide us toward a fully unified understanding of nature;
some even predict that all fundamental questions in physics will be solved
in the near future by pursuing these theoretical paths. On the other hand,
even supporters of these theories admit that their theoretical constructs
are often untestable in a definitive way, that they have had trouble with
most of the few falsification tests that have been identified, and that
they have been unable to anticipate major new observational discoveries.
This situation has been detailed elsewhere (Oldershaw, 1988) and will not
be repeated here in full, but let us briefly consider the current state
of affairs in cosmology. The Big Bang theory has always encountered serious
theoretical problems, such as the flatness problem, the smoothness problem,
and the horizon problem. These problems were "solved" by the
ad hoc addition of an Inflationary episode at about 10-35 sec
after the Big Bang, but this solution leads to other equally serious problems.
For example, the major prediction of the Inflated Big Bang theory is that
the matter density of the universe equals the critical density (i.e., Omega
= 1), but this prediction has been contradicted by most observationally
based estimates made to date (Rothman and Ellis, 1987). This theory also
leads to potential conflicts between the predicted age of the universe and
the estimated ages of its oldest constituents (Tayler, 1986). Moreover,
the Inflation scenario is totally dependent upon the validity of the GUTs
of particle physics, which are themselves beset by falsifications, arbitrariness,
and testability problems (Pickering, 1984). Even more worrisome is the
fact that the Big Bang theory failed to anticipate major empirical discoveries
of recent years, such as the large-scale inhomogeneity in the distribution
of matter, the large deviations from a smooth Hubble flow and, most importantly,
the dark matter constituting more than 90% of the matter of the universe
(Oldershaw, 1988). None of the variations on the Big Bang theme can provide
a convincing explanation for the existence of galaxies, and the Hubble constant
is uncertain by a factor of 2. In short, there is no justification for complacency
with regard to our current state of knowledge in the field of cosmology
(or particle physics).

For the past 10 years there has been a growing interest in
the "fractal" properties of nature's geometry, largely due to the
inspiration of Mandelbrot (1982). Fractal structures usually involve self-similarity,
a form of invariance with respect to transformations in scale, in which small
parts of a structure have geometrical properties that resemble the whole structure
or larger parts of the structure. A Russian doll of the type that has a doll
within a doll within a doll is a clearcut example of a self-similar structure.
Mandelbrot and many others who have followed his lead have identified examples
of self-similarity everywhere in nature: the clustering of galaxies, stars,
or atomic particles in a plasma; the branching of trees, rivers, or circulatory
systems; the cratering of astronomical bodies; the patterns of crystal growth;
the motions of turbulent fluids; the shapes of coastlines; the topology of
mountain ranges; etc. In fact it is difficult to think of any realm of nature
that does not include nontrivial examples of self-similarity. Although we
do not as yet have a fully satisfactory explanation for why self-similarity
should be so ubiquitous, we can unequivocally say that self-similarity is
one of nature's fundamental design properties. It seems reasonable and natural
to suspect that the solution to nature's biggest design problem, the design
of the cosmos itself, might involve self-similarity.

When a theory or paradigm is regarded as possessing beauty
or elegance, terms often associated with Einstein's General Theory of Relativity
or Darwin's Theory of Evolution, for example, it is meant that the theory
is conceptually simple and permits a pleasing unification of previously disjoint
facts or ideas about nature. As I hope to demonstrate in this paper, the
SSCM is conceptually very simple and proposes an unprecedented degree of unification
in the physics taking place on all scales of nature's hierarchy. Therefore,
the SSCM has the potential for being a remarkably beautiful theory; one scientist
(Sagan, 1980) has referred to the general notion of an infinite hierarchical
cosmos of self-similar systems as "one of the most exquisite conjectures
in science or religion." These potential attributes do not constitute
rigorous scientific support for the SSCM, but they do argue that it deserves
serious, open-minded consideration.

Another reason for exploring the possibility of a self-similar
cosmos is that it avoids at least three major philosophical problems that
have raised concerns about modern physics. Firstly, the Big Bang theory proposes
that the "initial state" of the entire universe was that of a singularity
with a radius of zero (space and time did not exist yet), but with infinite
pressure, temperature, and density. One might well ask how these latter quantities
could have any meaning without space-time. Moreover, the hypothetical initiation
of the expansion of the universe from a singular state represents an unexplained,
acausal event, a fact that is often downplayed in discussions of the Big Bang
theory, but which is obviously a theoretical drawback. The SSCM avoids this
problem by interpreting the large-scale expansion that we now observe as a
local phenomenon taking place in one particular metagalactic system on the
metagalactic scale of the hierarchy, much as stellar and galactic scale systems
can explode or undergo rapid expansion from a more compact state.

Secondly,
the Big Bang theory makes the extremely suspect assumption that nature's
hierarchy ends at about the scales where our observational capabilities
end, and that we just happen to find ourselves in the vicinity of the center
of that scale range. In the SSCM, on the other hand, there is no spectre
of an anthropocentric truncation of nature's hierarchy, since it postulates
that the hierarchy extends well beyond current observational limits, and
is perhaps completely unbounded.

Finally,
modem physics has something of a split personality in that the physics of
the microworld is hypothesized to be inherently different from the physics
of the macroworld, with a somewhat fuzzy interface between these two realms
wherein quantum microphysics rather mysteriously metamorphoses into classical
macrophysics. The SSCM hypothesizes that one set of physical laws holds
good for all scales of nature's hierarchy. Therefore, the fact that the
SSCM is not plagued by these three philosophical problems is another reason
for giving it due consideration.

The strongest argument for studying the concept of a self-similar
cosmos is the considerable amount of quantitative evidence that supports it.
In Section 3, 20 successful tests of the SSCM are discussed, and it is shown
that the match between theoretical predictions and observational estimates
far exceeds that expected by chance, or even that which could be achieved
by numerical "fudging." Compare this degree of empirical support
with that achieved by the highly regarded GUTs of particle physics (Oldershaw,
1988).

For these five reasons,
then, it would appear that the SSCM is worthy of serious attention in spite
of its divergence from generally accepted cosmological assumptions.

2. GENERAL DISCUSSION OF THE SELF-SIMILAR COSMOLOGICAL
MODEL

2.1. Heuristic Status

It should
be understood from the outset that the SSCM is still very much in the heuristic
stages of development. That is, the SSCM and the properties of nature which
led to the formation of its major hypotheses can be described in some detail
and the self-similar scaling equations that are the heart of the model can be
empirically derived and quantitatively tested, but the SSCM cannot as yet answer
the basic questions of why nature has a self-similar design and why the two
dimensionless constants of the scaling equations have the particular values
that are found empirically. The heuristic status of the SSCM may be viewed
as a shortcoming, but on the other hand it would seem to be unwise to risk stunting
the development of the SSCM by encasing the promising heuristic core in a hastily
constructed theoretical shell.

Because the
SSCM proposes a fundamentally different understanding of nature, and because
it would therefore significantly alter ideas in most branches of theoretical
physics, it is of considerable importance when studying this model that: (1)
all previous theoretical constructs should be regarded as being open
to question (a basic tenet of science), and (2) that observational data should
take primacy over theoretical assumptions or expectations when conflicts between
these two occur. Unfortunately, at present this is not always the case (Pickering,
1984).

2.2. Hierarchical Organization of the Cosmos

It is a self-evident
fact that nature has a nested hierarchical organization, though this fact is
often taken too much for granted. In our local planetary environment, for example,
"elementary particles" combine to form atoms, which are the building
blocks of molecules, which compose a vast array of macroscopic objects, which
are collected into planets, moons, asteroids, and comets, which are components
of the solar system. Taking a more cosmological perspective [in terms of the
ubiquity (Oldershaw, 1985) of the building blocks, the breadth of the spatial
domain under consideration, and the range of physical scale] it is known that
electrons, atomic nuclei, and ions are the primary building blocks of stars,
which are thought to be the primary building blocks of galaxies, which are clustered
into ever-larger aggregations until the limits of our observational abilities
are approached.

Considering
well-defined classes of relatively stable objects that have mass,
several major characteristics of the observable portion of the cosmological
hierarchy can be identified. The important question of the form of the dark
matter has been discussed previously (Oldershaw, 1986a,d) and will be a primary
topic of the sequel (Oldershaw, 1989c) to this paper.

As just mentioned, the cosmological hierarchy has a
nested organization wherein smaller objects are components of larger objects,
which, in turn, are components of even larger objects.

If we were to consider each class of objects as defining a
"level" in nature's hierarchical arrangement, and order these levels
in terms of increasing mass or range of masses associated with each class,
then the entire cosmological hierarchy involves a quasi-continuous
(Oldershaw, 1985) set of levels ranging in mass from about 10-27
g to at least 1045 g. Since building blocks at the levels of "elementary
particles" appear to have very discrete masses, a hierarchy that includes
them as major building blocks cannot be perfectly continuous in terms of the
range of masses of its constituents.

Although the overall mass range of objects comprising the
cosmological hierarchy is quasi-continuous, a relatively few classes of objects
account for large percentages of the mass of the observable universe, whereas
objects at most other levels account for infinitesimal percentages of the
observable mass (Oldershaw, 1986c). This means that the cosmological hierarchy
is highly stratified (Mandelbrot, 1982), and this concept is best illustrated
with some observational data.

At least 99.9% of all
observable mass is in the form of atomic and subatomic objects representing
the relatively narrow mass range of roughly 9 x 10-28 g for the
electron to about 9 x 10-23 g for an iron atom. This mass range
will tentatively be defined as the atomic scale of the cosmological hierarchy,
though the choice of cutoffs is somewhat arbitrary (Oldershaw, 1985). Between
the latter level and the levels at which stellar scale objects commence (about
1029 g) only a cosmologically insignificant amount of total mass
is in the form of the objects comprising the levels of this interscale range.
But at least 90% of the total observed mass is bound up in stellar and substellar
objects ranging in mass from about 10-5 M¤
to 8 M¤ (M¤
is one solar mass, or about 2 x 1033 g); this mass range will be
referred to as the stellar scale. Levels above the stellar scale do not involve
appreciable percentages of the cosmological mass until one reaches levels
designating galaxies, which range in mass from about 107 M¤,
to 1012 M¤ . At least
95% of all observable matter is bound up in objects populating these levels,
which are defined as the galactic scale of the cosmological hierarchy. These
three scales, involving about 15 orders of magnitude in mass out of a total
of about 72 orders of magnitude, dominate the observable portion of the cosmological
hierarchy, and in this sense the hierarchy can be regarded as quite stratified.
Moreover, there is evidence of further stratification within the atomic and
stellar scales. In terms of abundance, approximately 99% of all atoms are
accounted for by just two classes of atomic systems: hydrogen (in neutral
and H+ states) and helium (in neutral, He+, and He+2
states). Likewise on the stellar scale just two classes of stars, M dwarf
and K dwarf stars, account for about 99% of all observable stellar scale systems,
as will be discussed in Section 3. Whether a similar degree of stratification
occurs within the galactic scale cannot be determined at present due to uncertainty
in galactic mass estimates and limitations related to the available sample
of galaxies (see point 5 below). At any rate, the degree of stratification
characterizing the cosmological hierarchy is rather remarkable and worthy
of some reflection. From a cosmological perspective the observed portion
of nature is, to a rough first approximation, comprised of galaxies which,
when looked at “microscopically," are composed of (0.1-0.8) M¤
dwarf stars which, when looked at "microscopically," are composed
of hydrogen and helium. All of the other known classes of objects, most of
which populate interscale levels, are very minor components of the cosmological
hierarchy in the sense that when you add up the mass incorporated in any of
these classes and compare it to the total observed mass of the cosmos, the
resulting percentage is always less than 15%, and usually infinitesimal.

If it is assumed that the hierarchy has uppermost and lowermost
levels, then this is a purely theoretical assumption. Historically, whenever
observational capabilities have been significantly improved, new levels of
the cosmological hierarchy have been revealed. Currently the putative bounds
of nature's hierarchy are again tellingly close to the largest and smallest
size scales that can be adequately observed, and it would appear to be unwise
to regard the matter as being closed. The safest bet is that new levels of
nature's hierarchy remain to be discovered.

At face value it appears that the degree of stratification
might decrease above the galactic scale, and that the cosmological hierarchy
is asymmetric, since the atomic and stellar scales are separated by about
51 orders of magnitude in mass, whereas the stellar and galactic scales appear
to be separated by only about 7 orders of magnitude in mass. With regard
to the stratification question, the sample of observable galaxies is relatively
small, about 1011 galaxies as compared with about 1023
stellar objects and about 1080 subatomic particles. Moreover,
one must take into account the exceedingly small galactic scale "sampling
ratio," defined as the ratio of the radius of the observable portion
of the universe to the average radius of the systems of a particular scale.
The galactic scale sampling ratio is a paltry 105 as compared to
1017 for the equivalent stellar scale ratio and an awesome 1040
for the equivalent atomic scale ratio. Therefore, it is possible that only
a tiny sampling of the objects and phenomena occurring on size scales at and
above the galactic scale is currently available for scrutiny and this circumstance
is a serious hindrance to an accurate evaluation of the actual stratification
above the galactic scale. Concerning the issue of hierarchical asymmetry,
the SSCM proposes that the cosmological hierarchy is symmetric in all respects
and empirical tests presented in Section 3 seem to support the symmetry hypothesis.
This controversial issue has been discussed previously (Oldershaw, 1986a,d)
and will be given further attention in the sequel to this paper (Oldershaw,
1989c).

2.3. Discrete Self-Similarity

Given the
highly stratified organization of the observable portion of the cosmological
hierarchy, it seemed natural to compare atomic, stellar, and galactic scale
systems with regard to similarities and/or dissimilarities. Taking into account
the huge differences in spatio-temporal scale which tend to obscure inherent
similarities to a degree that is often seriously underestimated, I found that
there was a considerable potential for physically meaningful analogies among
atomic, stellar, and galactic scale systems. Let us consider several general
examples that suggest the possibility of interesting parallels between the physics
operating on different scales.

In the most general terms, typical systems from all three scales
involve distinct objects orbiting one another under the influence of attractive
(usually proportional to l/r2) "forces," e.g., atoms, the solar
system, and small galactic groups.

Relatively large-scale and highly collimated jets of matter,
often in a back-to-back configuration, are observed in both stellar and galactic
scale systems. The potential for meaningful physical analogies between these
jet phenomena has long been advocated by astrophysicists (Geldzahler et al,
1981; Geldzahler and Fomalont, 1986).

Likewise, astrophysicists have noted that in some ways, such
as their enormous densities, neutron stars are like stellar scale counterparts
to nuclear objects on the atomic scale. Moreover, atomic scale objects can
range in size from compact nucleons (r » 10-13 cm)
to relatively huge Rydberg atoms that are a billion times larger (up to r
» 10-4 cm). Similarly, stellar scale objects have
a size range of about a billion, from compact neutron stars (r < 106
cm) to stellar systems with radii of £ 1015 cm.

Binary spiral galaxy systems tend to avoid having parallel spins (Helou,
1984) and this phenomenon is also common to atomic scale systems. Additionally,
Tifft (1982) has presented data that are suggestive of quantization in the
orbital motions of binary galaxies.

As will be discussed in more detail in Section 3, both atomic
and stellar scale systems tend to have relationships between their angular
momenta (J) and masses (M) of the form J = kM2, where k is a constant.
Systems on both scales also tend to have relationships between their magnetic
dipole moments (m) and angular momenta of the
form m = DJ, where D
is a constant.

The remarkable potential for analogies between the solar system and an
atom in a highly excited state (Rydberg atom) has been known for some time
(Metcalf, 1980); in general the morphologies, kinematics, and dynamics of
Rydberg atoms and their stellar scale analogs are intriguingly similar (Oldershaw,
1982, 1986a, 1987b). For example, in both cases their radii and oscillation
periods are related by laws of the form of Kepler's third law, p2» KR3, where K is a constant (Oldershaw, 1989b).

Potential
analogies such as the six listed above, and others discussed in the SSCM references
cited above, emboldened me to consider the speculative hypothesis that atomic,
stellar, and galactic scale systems might be rigorously self-similar, i.e.,
that specific systems on a given cosmological scale have specific analogs on
all other cosmological scales, and that the properties of analogs from different
scales are quantitatively related by simple scale transformation equations.

The derivation
of a set of self-similar scale transformation equations, which can relate corresponding
length, time, and mass values for analog systems on different scales, was perhaps
the most important step toward quantitative testing of the cosmological self-similarity
hypothesis, since these equations would allow one to identify analogs on different
scales, to assess quantitatively their self-similarity, and to make definitive
predictions. From Mandelbrot's (1982) basic discussion of self-similarity,
a little physics (e.g., velocities should be scale invariant), and a knowledge
of the above-mentioned general properties of the cosmological hierarchy, one
can infer that the simplest scaling equations for a highly stratified self-similar
hierarchy would be:

Rψ = LRψ-1

(1)

Tψ = LTψ-1

(2) and

Mψ = LDMψ-1

(3)

where R, T, and M are length, time, and mass values pertaining to analog systems
on neighboring cosmological scales ψ and ψ-1,
and where L
and D are dimensionless scaling constants that must, for the present, be determined
empirically. The values ofL and
D are found to be approximately 5.2 x 1017 and 3.174, respectively,
and the methods by which these values were arrived at are discussed in Oldershaw
(1986a) and in the sequel (Oldershaw, 1989c) to this paper. In general,
these methods involve identifying a pair of putative analogs for which there
are reasonably accurate mass and radius estimates, and for which the analogy
seems dependable. Ratios
of analogous mass and radius measurements then yieldL
and D, since Rψ/Rψ-1 = L and
Mψ/Mψ-1 = LD.
The analog pair that was initially used consisted of the solar system, for
which accurate data are available, and a very highly excited Rydberg atom (n » 168),
an atomic scale system whose basic properties are both quantifiable and strongly
analogous to those of the solar system. The fact that L and
D are single-valued rather than multi-valued or continuous reflects the fact
that according to the SSCM, nature's hierarchy is modeled as having discrete
and symmetric stratification.

2.4. Summary of the Basic Model

The SSCM
views nature as a highly stratified, nested, and possibly unbounded hierarchy
of systems with atomic, stellar, and galactic scale systems comprising a discrete,
symmetric framework for the observable portion of the entire quasi-continuous
hierarchy. It is further hypothesized that the hierarchy is rigorously self-similar
such that radii, periods, masses, and in fact any corresponding parameters (Oldershaw,
1986a-e, 1987a) associated with analog systems on different scales are correlated
by the very simple scale transformations defined in equations (1)-(3). Given
the currently accepted theoretical models of atomic, stellar, and galactic systems,
one might be highly inclined to regard the latter hypothesis as being simply
impossible, i.e., of having no chance of applying to the real world. So much
more surprising, then, will be the results presented below of actual empirical
tests of the hypothesis. Nature, rather than human theoretical constructs,
should be the basis upon which we decide the merits or shortcomings of a scientific
hypothesis.

3. EMPIRICAL TESTS OF THE SELF-SIMILAR COSMOLOGICAL MODEL

3.1. Introductory Notes

Table I
(below)
presents the results of 20 retrodictive falsification tests of the SSCM. As
opposed to definitive predictions (Oldershaw, 1988), which predict unexpected
phenomena or the results of empirical experiments before they are known,
retrodictive falsification tests determine a theory's ability to "retrodict"
previously known data, i.e., they test a theory's consistency with observations.
Therefore, retrodictive falsification tests are inherently less stringent than
are tests involving definitive predictions. However, to the extent that a
theory can pass a large and diverse array of retrodictive falsification tests,
our confidence in the theory as a good approximation to natural phenomena
is commensurately increased. The final three tests listed in Table I come
reasonably close to being classified as true predictions, since they involve
relationships that were not thoroughly characterized prior to the tests;
several predictions of the SSCM that unquestionably meet the criteria for
definitive predictions have been presented before (Oldershaw 1986a,d; 1987a)
and will be discussed further in a forthcoming paper (Oldershaw, 1989c).

Table I. Retrodiction Tests of the SSCM

Test #

Test Parameter

Reference Parameter

Reference Value

Scale Factor

SSCP
Prediction

Observed

Value

1

M dwarf
abundance

H abundance

90 ± 2%

-

< » 90% >

< » 89% >

2

K dwarf
abundance

He abundance

9 ± 2%

-

< » 9% >

< » 10% >

3

Lower
limit RM dwarfs

Lower
limit R for H

1.6
x 10-8 cm

Λ

8.3
x 109 cm

8.7
x 109 cm

4

<R> for
white dwarfs

R for
He+

2.1
x 10-9 cm

Λ

1.1
x 109

0.9
x 109

5

Lower
limit R for white dwarfs

Lower
limit R for atomic ions

4.2
x 10-10 to
1.2
x 10-9 cm

Λ

2.2
x 108to
6.1
x 108 cm

5.5
x 108 cm

6

Range of R for
Main Seq. Stars

Range of R for
neutral atoms

1.6
x 10-8to
6.4
x 10-5cm

Λ

8.3
x 109to
3.3
x 1013cm

8.7
x 109to
3.4
x 1013 cm

7

Average
M for white dwarfs

Mass
of 4He

6.7
x 10-24 g

ΛD

1.14
x 1033 g

1.15
x 1033 g

8

Lower
M for white dwarfs

3He/4He
M ratio

0.75

-

8.7 x
1032 g

8.8 x
1032 g

9

Proton
radius

Schwarschild
R of black hole

Stellar
Scale G

ΛDΛ2/Λ3

0.81
x 10-13 cm

0.8
x 10-13 cm

10

Log
KS/KA from J=KiM2

-

-

Λ2/ΛD+1

-38.51

-38.41

11

Log ΔS/ΔA from μ=ΔJ

-

-

Λ0.5/Λ1.59

-19.31

-20.36

12

Typical
pulsar spin period

Typical
nuclear spin period

5
x 10-20 sec

Λ

0.03
sec

0.002 –
3.0
sec

13

R range
for galaxies

R range
for atomic nuclei

0.8
x 10-13 to
8.3
x 10-13 cm

Λ2

2.2
x 1022
to
2.2
x 1023 cm

0.9
x 1022 to
3.1
x 1023 cm

14

Typical
galaxy spin period

Typical
nuclear spin period

5
x 10-20 sec

Λ2

4.3
x 108 years

4.4
x 108 years

15

μ range
for neutron stars

μ range
for atomic nuclei

4.5
x 10-25to
1.8
x 10-23 G
cm3

Λ1.59 times Λ1.5

1030.34to
1031.94G
cm3

1030.3to
1031.3G
cm3

16

Period
range for He+

Period
range for white dwarfs

250-850
sec

Λ-1

4.8
x 10-16to
1.6
x 10-15sec

5.5
x 10-16to
1.6
x 10-15sec

17

Period
range for neutron stars

Period
range for atomic nuclei

1.3
x 10-22
to
7.8
x 10-21sec

Λ

6.8
x 10-5to
4.1
x 10-3sec

10.0
x 10-5to
1.2
x 10-3sec

18

Period-Radius
Law
for
stars

Period-Radius
Law
for
atoms

p2 = kar3

-

P2 =
KSR3

P2 = KSR3

19

KS values
from
#18

ka values
From
#18

1.6
x 10-7, 2.0 x 10-8sec2/cm3

Λ2/Λ3

3.0
x 10-25,
3.8
x 10-26sec2/cm3

3
x 10-25,
4
x 10-26sec2/cm3

20

Period
range
He
atoms
7 £ n £ 9

Period
range
RR
Lyrae stars

0.2 – 0.8
days

Λ-1

3.3
x 10-14to
1.3
x 10-13sec

» 5
x 10-14to» 1
x 10-13sec

Below I will
review each test and its results, referencing previous discussions of the test
and identifying new data that are applicable. Since all measurements involve
uncertainties, the reference, "predicted," and empirical values listed
in Table I are estimates and each should be thought of as being preceded by
an "approximately equals" symbol. Relevant sources and degrees of
uncertainty are discussed in the cited references and in this paper.

An important
caveat, already mentioned in Section 2, is that nature does not present us with
equivalent samples of atomic, stellar, and galactic scale systems. In terms
of numbers of systems and "sampling ratios" (see point 5 of Section
2.2), the values for the atomic, stellar, and galactic scales are about 1080,
1023 , and 1017, and 1040, 1017,
and 105, respectively. To put this into bold perspective, what
we observe of the galactic scale (Oldershaw, 1986d) is analogous
to studying the atomic scale on the basis of observing a mere 1011
subatomic particles crammed into a volume roughly comparable to that of a single
hydrogen atom. This sample would woefully under-represent the richness
of atomic scale phenomena, and therefore we must bear in mind that the available
galactic scale sample is similarly limited. The situation is quite a bit better
on the stellar scale, but the caveat against assuming equivalent samples
is still very important when making stellar-atomic comparisons.

The empirical
tests listed in Table I usually have the following format: a reference parameter
that has been measured with reasonable accuracy is identified for a class of
systems on a given scale, this value is then transformed according equations
(1)-(3) in order to yield a "predicted" counterpart value for the
analogous class of systems on a different scale, and finally the "predicted"
value is compared with empirical measurements made on the relevant class of
analog systems. Usually atomic scale systems are chosen as the source of reference
parameters because our empirical measurements of atomic scale parameters are
in general vastly superior to our quantification of stellar or galactic scale
parameters.

3.2. Discussion of Individual Tests

and

Since these two tests are intimately related, it will be convenient
to discuss them together. It has been firmly established (Trimble, 1975)
that the measured abundances (by numbers rather than by mass) of hydrogen
and helium are remarkably constant over a wide variety of cosmologically representative
samples: the sun's outer layers, the interstellar medium, meteorites, distant
stars, cosmic rays, and other galaxies. Hydrogen appears to account for 90±2%
of all atomic species, helium appears to make up 9±2% of all atomic species,
and elements heavier than helium only contribute about 1% to the total. Given
these atomic scale abundance values, the SSCM predicts that comparably representative
samples of large numbers of stellar scale systems will reveal that stellar
scale hydrogen and helium analogs account for approximately 90% and 9% of
the stellar scale sample, respectively. There is a technical difficulty with
a straightforward application of this test, but fortunately there is a way
to circumvent the problem. The atomic H and He abundances refer to total
H and total He; this means that atoms in neutral, partially ionized, and fully
ionized states are included in the atomic scale abundance determinations.
According to the SSCM, our present observational capabilities are not sufficient
to reliably detect stellar scale analogs to fully ionized atoms, i.e., bare
nuclei, and therefore fully ionized species must be excluded from the comparison
(Oldershaw, 1986c). If, instead of using total abundances, the abundances
of just neutral species are chosen for comparison, then on the atomic scale
the reference parameter values are essentially the same as for the total abundances,
and on the stellar scale all relevant counterparts are observable. By excluding
the partially ionzed species from the comparison, the serious complication
posed by widely differing ionization potentials is largely avoided.

The SSCM proposes (Oldershaw,
1986a) that stars with radii greater than about 9 x 109 cm, e.g.,
main sequence, giant, and supergiant stars, are stellar scale counterparts
to atoms in excited, but for the most part neutral, states. Equations (1)-(3)
predict that the stellar scale hydrogen analog has a mass of about 0.15M¤and
the helium analog has a mass that is four times larger, or about 0.6M¤.
As anticipated by the SSCM, recent data (Lupton et al., 1987; Low, 1985) show
a distinct abundance peak at about 0.62M¤
and a much larger peak that falls somewhere between 0.1M¤
and 0.2M¤. Since there is a considerable
amount of uncertainty involved in estimating stellar masses [note the broadness
of the abundance peaks of Lupton et al. (1987)], the stellar scale abundances
of H and He analogs will be defined here as the abundances of stars with masses
estimated to be in the ranges 0.1M¤
to 0.4M¤ and 0.45M¤
to 0.75M¤, respectively. These mass
ranges correspond quite well to the estimated mass ranges of M dwarf and K
dwarf stars, and therefore the SSCM predicts that the abundances of M dwarf
and K dwarf stars should be about 90% and 9%, respectively. Quantitative
determinations of these stellar abundances are exceedingly hard to find in
the literature, but Wood (1966) has made a comprehensive attempt, and for
his most reliable sample of galaxies the M dwarf abundance ranges from 81%
to 95% with an average of 89%, While the K dwarf abundance ranges from 6%
to 18% with an average value of 10%. The average values are quite close to
the predicted values.

Since M dwarf stars are identified with stellar scale analogs
to hydrogen in neutral but usually excited states, one can take the ground-state
radius for H, scale it according to equation (1), and arrive at a SSCM prediction
for the lower limit radius of an M dwarf star. The only difficulty here is
that neither an atom nor a star has a distinct boundary at a fixed radius,
but rather both have somewhat ephemeral boundaries. The radius encompassing
90% of the electronic charge distribution was chosen as an appropriate estimate
for the ground-state radius of H, and the consequent prediction for the lower
limit radii of M dwarf stars (8.3 x 109 cm) was found to be in
good agreement with observational estimates of approximately 8.7 x 109
cm (Oldershaw, 1986a).

Their masses, abundances, and probable origins in planetary
nebulae all serve to identify (Oldershaw, 1986a,c) the overwhelming majority
of white dwarf stars as stellar scale analogs to He+ ions in their
ground states. Therefore the estimated radius for a ground state He+
ion (roughly 0.4ao, where ao is the Bohr radius) can
be scaled according to equation (1) to yield a predicted average radius for
white dwarf stars. The resulting prediction of roughly 1.1x109
cm is found to be in reasonable agreement with the observationally estimated
average radius (Greenstein, 1985) of 0.9 x 109 cm for white dwarf
stars, given the uncertainties associated with the latter value. Parenthetically,
it has been noted (Oldershaw, 1982) that the morphologies of the shells being
ejected in planetary nebula systems, the cores of which are interpreted as
predominantly He+ analogs, are intriguingly similar to the morphologies
of electronic wave functions in atoms.

Although nearly all white dwarf stars have been identified
as He+ analogs, with masses of approximately 0.45M¤
(see test 8 below) and 0.60M¤, very
small numbers of stellar scale analogs to more massive ions are expected to
be found in this class of objects. It should be clarified that according
to the SSCM the class of white dwarf stars is analogous to the class of highly
(but not fully) ionized atomic scale ions with remaining electrons populating
very low energy levels. Therefore, if one scales the lower limit radius for
atomic ions with masses greater than four atomic mass units according to equation
(1), then one should arrive at an SSCM prediction for the lower limit radius
for a white dwarf star (Oldershaw, 1986a). Ionic radii (Weast, 1971-1972)
for almost fully ionized ions more massive than He have a lower limit value
of roughly 0.08ao, where ao is the Bohr radius, and
singly ionized ions have a lower limit radius of about 0.22ao.
It is not entirely clear which lower limit represents the better reference
parameter for this test, and so we will only expect that the observed lower
limit radius for white dwarf stars (or better, for those whose radii have
been estimated so far) will be in the range (L)(0.08ao)
to (L)(0.22ao), or 2.2 x 108 cm to 6.1x108
cm. This is found to be the case; the observed lower limit is about 5.5x108
cm (Greenstein, 1985).

As noted above, the SSCM identifies main sequence, giant,
and supergiant stars as stellar scale analogs to excited, but primarily neutral,
atoms. The latter have a large range of radii extending from approximately
3ao for the ground state of H to an approximate radius of 12,100ao
for the largest Rydberg atoms that are commonly observed (Percival, 1980).
A coarse but useful test of the SSCM can be achieved by using equation (1)
to scale the limits of this range up to stellar scale values, and to inquire
whether this predicted range corresponds to the observed radius range for
"normal" stars. Results of this test are in good agreement with
expectations of the SSCM, since the lower limit radius for M dwarf stars is
roughly 3Ao, where Ao is the stellar scale equivalent
to the Bohr radius, and supergiant stars have observed radii up to roughly
12,14OAo (de Vaucouleurs, 1970).

Having determined that the stellar scale equivalent to the
mass of the hydrogen atom is approximately 0.15M¤
(Oldershaw, 1986a), the SSCM predicts that the stellar scale analogs to helium,
i.e., K dwarfs and most white dwarfs, will be about four times more massive,
or 0.60M¤. Observational results
confirm that 0.6M¤ is an excellent
estimate for the average mass of K dwarfs (Lupton et al., 1987), and the distribution
of masses for white dwarf stars is a surprisingly narrow peak centered on
about 0.6M¤ (Schoenberner, 1981;
Mallik, 1985). The nuclei of planetary nebulae, which are also identified
as He+ analogs and precursors of white dwarf stars, have a remarkably
sharp mass distribution centered on 0.58M¤
(Schoenberner, 1981). The sequel (Oldershaw, 1989c) to this paper will contain
a discussion of the SSCM prediction that the actual distribution of stellar
masses is much more discrete than is inferred at present; the very sharp mass
distribution for the nuclei of planetary nebulae is encouraging evidence along
these lines.

If white dwarf stars are predominantly analogs to helium ions,
then they must be primarily analogs to 4He+ ions, which
are by far the most common isotope of helium. However, one would expect very
small numbers of 3He+ analogs to be included in the
present sample of white dwarf stars, and therefore one would predict that
the lower limit mass for white dwarf stars is approximately (3/4)(0.58M¤)
= 0.44M¤. Two observational estimates
(Mallik, 1985; Greenstein, 1985) of this parameter are 0.45M¤
and 0.44M¤.

Using equations (1) and (3), one may calculate (Oldershaw,
1986a) that the mass and radius of the stellar scale analog to the proton
are approximately 0. 145 M¤ and 0.42
x 105 cm, respectively. It was immediately noticed that the radius
for the stellar scale proton analog is very close to the Schwarschild radius
(0.428 x 105 cm) for a stellar object with a mass equal to 0.145M¤.
According to the SSCM, therefore, the radius of the proton should be equal
to the Schwarschild radius for an object with a mass equal to 1.67 x 10-24
g, if the "constants" in the Schwarschild radius equation:

RS = 2GψM/c2

(4)

are scaled to their proper atomic scale values (Oldershaw, 1986a,de). The value
for the velocity of light c is invariant with respect to scale transformations,
but the Newtonian gravitational constant G has dimensions L3/M
T2, and therefore according to equations (1)-(3) the atomic scale
value G-1 is L2.174 times
larger than the stellar scale value Go . Solving equation (4) with
M = 1.67 x 10-24 g and G-1 = (L2.174)
(6.68 x10-8 cm3/g sec2) gives a predicted
radius of 0.81 x 10-13cm; the empirically estimated
radius for the charge distribution of the proton (Bethe and Salpeter, 1957)
is approximately 0.8x10-13 cm.

and

These two tests have been presented in detail before (Oldershaw,
1986b) and here I will only outline the rationale for the tests and repeat
the results. It has been observed that many stellar scale systems have a
relationship between their masses M and angular momenta J of the form J =
KsM2, where Ks is a constant with dimensions
L2/ MT. Similarly, it has been observed that families of atomic
scale systems obey relationships of the form J = Ka M2.
According to the scaling rules of the SSCM, the logarithm of Ks/Ka
should have a value of approximately -38.51, and this is in good agreement
with the rough empirical estimate of -38.41 (± 3.50). Likewise,
both stellar scale and atomic scale systems tend to have a relationship between
their magnetic dipole moments m and angular momenta J of the
form m = DJ. The scaling rules of the SSCM lead
to the expectation that the logarithm of Ds/Da
should be about -19.31, which can be compared with the empirical estimate
of -20.36 (± 2.43). Because of the very large error bars on the
empirical values for Ks/Ka, and Ds/Da
,these tests only show that the SSCM predictions for Ks/Ka
and Ds/Da are "in the
right ballpark." Perhaps in the future, improved empirical constraints
will permit more stringent versions of these tests.

A classical spin period for a typical atomic scale nucleus
is estimated to be about 5x10-20 sec (Oldershaw, 1986d). Since
the SSCM identifies neutron stars as stellar scale analogs to atomic nuclei,
equation (2) can be used to scale up the nuclear spin period of 5x10-20
sec to an expected value of approximately 0.03 sec for the spin period of
a typical neutron star. To date, the observed range of spin periods for pulsars
is 0.002 sec to about 3.0 sec, and therefore the predicted spin period does
fall within the empirical range for neutron stars. On the other hand, it
should be mentioned that pulsar spin periods in the range 0.1 to 1.0 sec are
far more common in present samples than those below 0.1 sec. An intriguing
phenomenon that both atomic nuclei and pulsars share is that of abrupt "glitches"
(Stephens, 1985) wherein the spin frequency of the system seemingly instantaneously
goes from regular decrease to a significantly higher value, and then resumes
a slow decrease from the higher frequency. The observed pulsar "glitches"
(Lyne, 1987) have so far involved only very small frequency jumps (Df/f
£ 10-6) as compared with the very large
"glitches" (Df/f on the order of 10-1) seen
in atomic nuclei, but the analogy is an interesting one and perhaps comparably
large pulsar "glitches" will be observed in the future. To date,
only 14 "glitches" in 7 pulsars have been observed, but statistics
indicate that they should be a very common phenomenon.

If equations (1)-(3) do relate self-similar phenomena on different
scales of the cosmological hierarchy, then the high-velocity (average value ³400
km/sec) random motions of galaxies unambiguously require that their analogs
on the atomic scale are atomic nuclei under fully ionized plasma conditions
(Oldershaw, 1986d). Therefore, if the range of radii for atomic nuclei, which
is about 0.8x10-13 cm to 8.3x10-13 cm, is scaled up
to galactic scale values according to equation (1), i.e., multiplied by L2
, then the resulting range of about 2.2x1022 to 2.2x1023
cm should compare favorably with the empirically estimated range for the radii
of galaxies. There is a significant amount of uncertainty in galactic radius
estimates, primarily because the Hubble constant is uncertain by a factor
of 2 and the exact extent of dark matter halos of galaxies is often difficult
to estimate. However, the smallest galaxy for which the dark matter halo
has been taken into account (Kormendy, 1985) has a radius of ³
0.9x1022 cm and the largest galaxies (Saslaw, 1985) have radii
of roughly 3.1x1023 cm. The agreement between the predicted and
the empirically estimated radius ranges is quite good considering the present
observational uncertainties, and a more exact correspondence is a viable possibility.
Both galaxies and atomic nuclei have shapes that are well-represented by McClaurin
spheroids and Jacobi ellipsoids, including prolate and triaxial shapes (Oldershaw,
1986d).

In test 12 a typical spin period for an atomic nucleus and
the range of spin periods for pulsars were shown to be correlated in a manner
that was consistent with the SSCM predictions. A further SSCM prediction
is that multiplying the atomic scale spin period of about 5x10-20
sec by L2, in accordance with equation (2), should
yield a typical galactic spin period. The numerical value is about 4.3x108
years, and this spin period is approximately equal to a rough estimate (Mihalas
and Binney, 1981) of the spin period of 4.4 (±2.2) x 108
years for our galaxy, which is in all respects a typical galaxy.

As mentioned above, the SSCM unambiguously identifies atomic
scale nuclei and stellar scale neutron stars as self-similar analogs. Therefore
the range of magnetic dipole moments m, with dimensions of M1/2L3/2,
for atomic nuclei should be related to the m range for neutron
stars by a scaling factor of (LD/2)(L3/2)
= 4.9x1054 . The range of m values for cosmologically
abundant atomic nuclei (Oldershaw, 1987a) is 4.5x10-25 to 1.8x10-23
G cm3, and so the predicted range for neutron stars is roughly
1030.34 to 1031.94 G cm3. The estimated
range of m values for neutron stars is roughly 1030.3 to
1031.3 G cm3 (Kundt, 1986), which is in good agreement
with the predicted range, given the theoretical and empirical uncertainties
involved in this test.

Since the majority of white dwarf stars have been identified
as self-similar analogs to He+ ions, and since white dwarfs appear
to have preferred oscillation periods (Wesemael et al, 1986) of approximately
250 (±100) and 850 (±100) sec, it can be predicted via equation
(2) that He+ ions should have major transition periods of about
(250 sec)(L-1) = 4.8x10-16 sec and (850
sec)(L-1) = 1.6x10-15 sec. In fact, these
predicted periods are in good agreement with two of the three major transiton
periods of He+ ions: 5.5x10-16 and 1.6x10-15
sec (Oldershaw, 1989a).

Since neutron stars have been identified as self-similar counterparts
to atomic scale nuclei (Oldershaw, 1986a), the SSCM predicts that the ranges
of vibrational periods for these two classes of systems should be correlated
by equation (2). Vibration periods in atomic nuclei range from about 1.3x10-22
to 7.8x10-21 sec, and therefore the anticipated range of vibration
periods for neutron stars should be approximately 6.8x10-5 to 4.1x10-3
sec (Oldershaw, 1989a). This predicted range is in good agreement with the
empirically determined range of 10.0x10-5 to 1.2x10-3
sec, considering that the latter range is based on a very small sample size
(Carroll et al, 1986).

and

The SSCM identifies the majority of main sequence, giant,
and supergiant stars as stellar scale analogs to neutral atoms in highly excited
Rydberg states (Oldershaw, 1986a, 1987b). It also predicts that any well-defined
physical phenomenon observed on either the atomic or stellar scale will have
an analogous counterpart on the other scale. When Rydberg atoms undergo transitions
to lower energy states they oscillate
with periods p that are related to the average radii r in the following manner
(Percival, 1980):

p2»
k1r3

(for l » n)

(5)

p2»
k2r3

(for l << n)

(6)

where n is the principal quantum number, l is the azimuthal quantum number,
k, is a constant equal to (p0)2/(a0)3,
and k2 is a constant equal to (p0)2/(2a0)3
. The parameter p0 is the minimum transition period for hydrogen
and a0 is the Bohr radius. From the fact that Rydberg atoms
obey approximate relationships of the form of Kepler's third law, i.e.,
p2» kir3,
it can be predicted that variable stars with radii ³
lR¤, which have been identified
as stellar scale analogs to Rydberg atoms undergoing transitions to lower
energy states (Oldershaw, 1987b), will have periods P and radii R that obey
approximate relationships of the form P2»
KiR3, where the Ki represent analogs to
the ki. It has been demonstrated (Oldershaw, 1989b) that a wide
variety of variable stars, including delta Scuti, RR Lyrae, beta Cepheid,
classical Cepheid, and supergiant variables, do indeed obey period-radius
relations of the predicted form. Moreover, it has been shown that the atomic
scale constants k1 and k2 are quantitatively related
to their stellar scale counterparts K1 and K2 by the
self-similar scaling rules embodied in equations (1)-(3). A third P-R relationship
with a K3 value that is closely related to K1 and
K2 has been identified for variable stars and has led to the
prediction that an analogous p-r relation will be found for a subset of
Rydberg atoms (Oldershaw, 1989b).

This final test of the SSCM requires more discussion than
its predecessors because it has not been published previously. The proposed
analogy between variable stars and Rydberg atoms undergoing transitions leads
to the expectation that the periods of variable stars have quantized values,
as is the case with their atomic scale analogs. This expectation will be
explored below, but several important caveats must be mentioned first. When
atomic scale quantization is observed, one of two general strategies has been
employed: strategy A is to observe a perfectly homogeneous sample of atoms
under rigorously controlled ambient conditions. When strategy A is not feasible
because one cannot regulate the homogeneity of the atomic species and/or the
ambient physical conditions, then strategy B is to sample enormous numbers
(³ 1020) of atoms in the hope that
discrete peaks will rise above the nearly continuous background. On the stellar
scale one is faced with the following observational circumstances.

(b) Values
of n for individual stars can range from 1 to at least 100.

(c) For each
value of n there are n different energy levels due to orbital angular momentum
considerations, i.e., l can vary from 0 to n - l for each value of n.

(d) If spin
considerations are included, then the above-mentioned energy levels are
further split into an even larger set of levels.

(e) Since
the energy levels of Rydberg atoms can be significantly shifted by ambient
electric and magnetic fields, the SSCM asserts that an analogous shifting
of energy levels can also occur in the case of their stellar scale counterparts.
Variable stars from different locations within our galaxy, i.e., near the
nucleus, in the outer halo, in the spiral arms, or in globular clusters,
would therefore be expected to have period distributions that are influenced
by differing galactic scale electromagnetic environments.

If careful thought is given to these five considerations, which would
serve to generate a dense "forest" of transition periods for Rydberg
atoms or their analogs, then it is clear that expecting to find textbook-style
evidence for quantized periods among variable stars, based on a maximum
sample size on the order of 104 periods, is essentially ruled
out at present. Under the existing observational circumstances even less
overt evidence for quantization in atoms or variable stars would still be
very difficult to obtain, since strategy B is precluded by having only a
tiny sample of systems and since strategy A is hampered by our inability
to manipulate the sample or the ambient physical conditions that affect
the sample. However, all is not lost. Granted that blatant examples of
quantization are not to be expected, one might still hope to observe less
overt evidence of quantization in the following manner. Since the sample
size is invariably going to be small, the best strategy is to identify as
homogeneous a subsample of variable stars as possible, with the hope of
minimizing the number of different species, the spread of n and l values,
and the influence of differing ambient physical conditions.

RR Lyrae stars constitute perhaps the best
candidate for a class of variable stars that meets the desired criteria.
Their masses are found to cluster around 0.6M¤
(Stothers, 1981) and therefore the SSCM unambiguously identifies them as
primarily helium analogs. The overwelming majority of their radii fall
within the range of 4R¤ to 7R¤
(Stothers, 1981), and from this fact the SSCM identifies (Oldershaw, 1987b)
the range of n values for RR Lyrae variables as n = 7 to n = 9. Also, their
position on a period-radius graph shows that they represent the l <<
n case (Oldershaw, 1989b); here we will assume that l £ 2.
Therefore, if reasonably large samples of RR Lyrae variables from reasonably
homogeneous galactic environments are analyzed in terms of relative frequencies
of oscillation periods, then the SSCM anticipates that evidence for discrete,
preferred periods will be present, though the statistical significance might
be low. The range of the period distribution and the preferred periods
for the RR Lyrae stars should be correlated with corresponding He transition
periods in a manner consistent with equation (2).

I have investigated the period distributions
for several RR Lyrae subsamples taken from the General Catalogue of Variable
Stars [the Third Edition and its Supplements] (Kukarkin et al, 1969-1970),
and two useful empirical findings have resulted from these investigations.
First, about 99% of the RR Lyrae stars have periods in the range 0.2 - 0.8
days. Second, there tend to be recurrent peaks in the subsample period distributions
at periods of 0.32 ± 0.01, 0.37 ± 0.01, 0.40 ± 0.01, 0.44 ±
0.01, 0.47 ± 0.01, and 0.52 ± 0.01 days. The strengths of these
preferred periods varies from subsample to subsample and their position
is sometimes shifted by ± 0.01 day, but their recurrence in different subsamples
lends credibility to the hypothesis that RR Lyrae variables have preferred
periods. As an example, Table II presents the distribution
of periods for a subsample of 672 RR Lyrae variables that were listed for
the Sagittarius region in the Second Supplement to the Third Edition
of the General Catalogue of Variable Stars (Kukarkin et al, 1974).
This typical subsample shows that the range of 0.2-0.8 days includes almost
all of the observed periods and it has what appears to be distinct peaks
at the six values listed above.

Table
II.Distribution of Periods for a Sample of 672 RR
Lyrae Variables and Relevant

Scaled Transition Periods for H, He, and Li Atoms

RR Lyrae period distribution

Scaled atomic periods

DP (days)

N

DP (days)

N

H

He

Li

0.150-0.159

1

0.500-0.509

15

0.383

0.276

0.250

0.160-0.169

0

0.510-0.519

32ß

0.558

0.287

0.256

0.170-0.179

2

0.520-0.529

35

0.323

0.362

0.180-0.189

1

0.530-0.539

35

0.326

0.369

0.190-0.199

0

0.540-0.549

30

0.354

0.378

0.200-0.209

1

0.550-0.559

23

0.378

0.398

0.210-0.219

0

0.560-0.569

27

0.389

0.529

0.220-0.229

2

0.570-0.579

22

0.404

0.543

0.230-0.239

4

0.580-0.589

18

0.406

0.579

0.240-0.249

2

0.590-0.599

13

0.424

0.590

0.250-0.259

2

0.600-0.609

14

0.432

0.798

0.260-0.269

5

0.610-0.619

9

0.440

0.866

0.270-0.279

5

0.620-0.629

7

0.464

0.280-0.289

5

0.630-0.639

11

0.474

0.290-0.299

7

0.640-0.649

10

0.478

0.300-0.309

4

0.650-0.659

9

0.513

0.310-0.319

4

0.660-0.669

4

0.518

0.320-0.329

11ß

0.670-0.679

0

0.552

0.330-0.339

10

0.680-0.689

2

0.565

0.340-0.349

8

0.690-0.699

6

0.590

0.350-0.359

8

0.700-0.709

3

0.633

0.360-0.369

12

0.710-0.719

3

0.645

0.370-0.379

14ß

0.720-0.729

2

0.681

0.380-0.389

6

0.730-0.739

0

0.752

0.390-0.399

5

0.740-0.749

1

0.400-0.409

16ß

0.750-0.759

1

0.410-0.419

4

0.760-0.769

0

0.420-0.429

7

0.770-0.779

1

0.430-0.439

16

0.780-0.789

0

0.440-0.449

28ß

0.790-0.799

1

0.450-0.459

27

0.800-0.809

2

0.460-0.469

24

0.810-0.819

0

0.470-0.479

43ß

0.820-0.829

1

0.480-0.489

25

0.830-0.839

0

0.490-0.499

26

0.840-0.849

0

Assuming that RR Lyrae variables correspond to the case 7 £
n £ 9, l £ 2, and <n2 - n1>
= 1, one can compare the stellar scale results with the corresponding transition
period data for He, including singlet and triplet systems. For ease of comparison
the He periods are scaled up to units of "days," i.e., multiplied
by a factor of L in accordance with equation (2). The scaled
He periods are determined by the calculation

P = ( h /DE)(L)

(7)

where P is the transition period scaled up to "days,"
DE is the energy level separation (Bashkin and Stoner,
1975),Lis the scaling constant from
equations (1)-(3), and h is Planck's constant. The resulting transition period
data for He are given in Table II. The range of relevant
transition periods for He is 0.276 to 0.752 "days," which is in reasonable
agreement with the predicted range of 0.2 to 0.8 "days." And as predicted,
the set of 24 transition periods for He contains counterparts to each of the
six preferred periods identified in the RR Lyrae sample. In order to test the
uniqueness of the correspondence between the He periods and the preferred RR
Lyrae periods, the same calculations were undertaken for hydrogen and lithium
atoms, and the relevant transition periods for these atoms are also listed in
Table II. There is no correlation between the H periods
and the preferred RR Lyrae periods. In the case of Li, three of the preferred
RR Lyrae periods, including the largest and most discrete peak, have no counterparts
among the scaled Li transition periods. Thus, the correspondence between the
He and RR Lyrae periods appears to be significant.

This first attempt to investigate the possibility of quantization
in the periods of variable stars represents a very approximate test that involves
a small sample size and relies on numerous assumptions. Yet the results are
reasonably encouraging and they suggest the way to achieve more rigorous quantization
tests in the future (Oldershaw, 1989b). Such tests would require much larger
samples of variable stars that have been segregated according to galactic location,
and they would require highly accurate period, radius, and mass data. It would
also be interesting to test whether the relative peak heights of the preferred
periods for variable stars match up with the transition probabilities for the
corresponding periods of the atomic scale analogs.

3.3. Implications of the Empirical Tests

There are only three plausible explanations for the general agreement
between the predictions and the empirical data in the 20 tests discussed above:
chance, fudging of various types, or cosmological self-similarity. A rough
and very conservative calculation of the probability that the agreement could
have resulted by chance can be made in the following manner. Assume that the
probability of a chance agreement for each test is £
1/3, i.e., the prediction could be unacceptably high, unacceptably low, or within
the error bars of the relevant empirical parameter. Then the maximum probability
that the same set of scaling rules could pass 20 such tests by chance is (1/3
)20 or one chance in 3,486,784,424 tries, which is to say that chance
would be an extremely unscientific explanation for the favorable results presented
above.

Fudging, or arbitrarily adjusting a theory so that it comes into
agreement with observational data, has always been and still is (Oldershaw,
1988) a standard tool of the theoretician, though one that tends to be used
furtively. The question to be considered here is whether fudging could account
for the apparent success of the SSCM and its scale transformation equations.
If one could make arbitrary choices with regard to the proposed analog pairs,
the form of the scaling equations, and/or the values of the constants appearing
in the scaling equations, then could the arbitrarily fudged theory pass the
20 falsification tests presented above, or a comparable set, even though nature
was not fundamentally self-similar? The author's answer to this question, after
investigating such matters for over 10 years, is that if self-similarity was
not a global property of nature, then a fudged theory that could pass these
particular falsification tests, or equally fundamental ones, would be hopelessly
complicated and arbitrary. Moreover, as one tried to test the theory beyond
the data that it was constructed around, it would quickly fail. In contrast,
the SSCM has very simple conceptual foundations and scaling equations, the identifications
of analog pairs are always based on two or more fundamental properties such
as mass, radius, and spin period, and nearly half of the tests (numbers 10-12
and 14-20) were conceived and conducted after the theoretical foundations of
the SSCM, its major analog pair identities, the form of the scaling equations,
and the values of L and D had been submitted for publication (Oldershaw,
1986a). In the absence of a convincing demonstration to the contrary, for example,
a demonstration that an equally simple and successful alternative to the SSCM
can be arbitrarily constructed, the fudging explanation is scientifically untenable.
The number, diversity, and fundamental nature of the quantitative falsification
tests passed by the SSCM strongly support the contention that nature manifests
discrete cosmological self-similarity and that equations (1)-(3) uniquely relate
the physical properties of atomic, stellar, and galactic scale systems.

4. CONCLUSIONS

In this paper the general concepts, and the self-similar scale transformation
equations, of the SSCM have been discussed, and 20 successful tests have been
presented. The simplicity of this model and its ability to quantitatively
relate atomic, stellar, and galactic scale phenomena suggest that a new property
of nature has been identified: discrete cosmological self-similarity. Although
the SSCM is still in the early heuristic stage of development, it may be the
initial step toward a truly remarkable unification of our considerable, but
fragmented, physical knowledge. Major questions yet to be answered concern
the exactness of the cosmological self-similarity (i.e., is the self-similarity
accurate only to a factor of about 2, or is it exact?) and the number of scales
in the cosmological hierarchy (i.e., finite or infinite?). It has been argued
previously that these two questions are interrelated (Oldershaw, 1981b); for
example, exact self-similarity necessitates an infinite hierarchy. An even
more fundamental question is: why should nature be globally self-similar and
rife with examples of local self-similarity?

A forthcoming paper (Oldershaw, 1989c) on the SSCM will discuss
the paradigm in more technical detail. It will also review several definitive
predictions by which the SSCM can be put to very rigorous tests, and it will
discuss major unresolved problems that raise doubts about some aspects of the
model. The review will conclude with a discussion of the diverse implications
of the SSCM.